Axioms for the optimal stable rules and fair-division rules in a multiple-partners job market

In the multiple-partners job market, introduced in (Sotomayor, 1992), each firm canhire several workers and each worker can be hired by several firms, up to a given quota. Weshow that, in contrast to what happens in the simple assignment game, in this extension, thefirms-optimal stable rules are neither valuation monotonic nor pairwise monotonic. However,we show that the firms-optimal stable rules satisfy a weaker property, what we call firmcovariance,and that this property characterizes these rules among all stable rules. This propertyallows us to shed some light on how firms can (and cannot) manipulate the firms-optimal stablerules. In particular, we show that firms cannot manipulate them by constantly over-reportingtheir valuations. Analogous results hold when focusing on the workers. Finally, we extend tothe multiple-partners market a known characterization of the fair-division rules on the domainof simple assignment games